Quadratic Jordan algebra

Definition

Let $\Phi$ be a commutative ring with unit, and let $J$ be a module over $\Phi$ with an element $1 \in J$. Let $U \colon J \to \End_\Phi(J)$, $x \mapsto U_x$, be a quadratic map: $U_{\alpha x} = \alpha^2 U_x$ for $\alpha \in \Phi$, and $(x, z) \mapsto U_{x + z} - U_x - U_z$ is bilinear. For $x, y, z \in J$ put $\set{x, y, z} = (U_{x + z} - U_x - U_z)\, y$, and let $V_{x, y}$ be the map $z \mapsto \set{x, y, z}$. The triple $(J, U, 1)$ is a quadratic Jordan algebra if for all $x, y \in J$:

1. $U_1 = \id_J$;
2. $U_x V_{y, x} = V_{x, y} U_x$;
3. $U_{U_x y} = U_x U_y U_x$;
4. clauses 1 to 3 also hold after extending scalars to $J \otimes_\Phi \Omega$, for every commutative $\Phi$-algebra $\Omega$ with unit.